Sun toughness and path-factor uniform graphs

School of Mathematics and Information Sciences, Yantai University Yantai, Shandong 264005, P.R. China

^* Corresponding author: liuhongxia@ytu.edu.cn

Received: 7 October 2022
Accepted: 11 November 2022

Abstract

A path-factor is a spanning subgraph F of G such that each component of F is a path of order at least two. Let k be an integer with k ≥ 2. A P_≥k-factor is a spanning subgraph of G whose components are paths of order at least k. A graph G is called a P_≥k-factor covered graph if for any edge e of G, G admits a P_≥k-factor covering e. A graph G is called a P_≥k-factor uniform graph if for any two distinct edges e₁ and e₂ of G, G has a P_≥k-factor covering e₁ and excluding e₂. In this article, we claim that (1) a 4-edge-connected graph G is a P_≥3-factor uniform graph if its sun toughness s(G) ≥ 1; (2) a 4-connected graph G is a P_≥3-factor uniform graph if its sun toughness s(G)>4/5.